Numerical Errors

A numerical error is either of two kinds of error in a calculation. The first (a rounding error) is caused by the finite precision of computations involving floating-point values. Increasing the number of digits allowed in a representation reduces the magnitude of possible roundoff errors, but any representation limited to finitely many digits will still cause some degree of roundoff error for uncountably many real numbers.

The second type of error (sometimes called the truncation error) is the difference between the exact mathematical solution and the approximate solution. Suppose, that we have defined an equidistant mesh $\lbrace x_i\rbrace$ and let us consider first a local error which arises from only one step of some numerical scheme.

A difference

$$\boxed{ \varepsilon_{i+1}=u(x_{i+1})-u_{i+1}}$$ (1.16)

is said to be a local discretization error in the point $x_{i+1}$ . Here $u(x_{i+1})$ is a exact solution of the problem in the point $x_{i+1}$ whereas $u_{i+1}$ describes a value in this point, calculated using the numerical approximation. In other words, the local discretization error can be interpreted as a residuum, if one put the numerical solution into the exact one. Now, if we put the Taylor expansion in the vicinity of the point $(x_i, u(x_i))$ into the equation of interest, one get the information how fast the local error tends to zero with the spacing $\triangle x$ . This observation leads to the definition of the so-called consistency order:

One says, that a numerical scheme possess a consistency order $p$ , if

$$\vert\varepsilon_{i+1}\vert\leq C\triangle x^{p+1}, \quad i=0,1,2,\ldots,$$ (1.17)

where $C$ is a constant.

As mentioned above, the local error gives information about the accuracy of the numerial scheme, i.e., about the error in one its step. At the end of calculation one can calculate an accumulated or a global discretization error in tghe point $x_{i+1}$ :

$$\boxed{e_{i+1}=u(x_{i+1})-u_{i+1}.}$$ (1.18)

The value of the global error gives information about convergence of the approximation to the exact solution of the problem if the spacing value $\triangle x$ tends to zero, i.e.,

A numerical scheme is said to be convergent, if for the global error $e_i$ one can write

$$\max_{i=1\ldots n}\vert e_i\vert\rightarrow 0 \, \triangle x\rightarrow 0.$$ (1.19)

The scheme posseses a convergence order $p$ , if

$$\max_{i=1\ldots n}\vert e_i\vert\leq C\triangle x^p,$$ (1.20)

where $C$ is a constant.

Notice: At a first glance the global error tends to zero with the decreasing of $\triangle x$ , so the mesh should be refined. However, decreasing of $\triangle x$ leads to the increasing of the rounding error. Another point to emphasize is that decreasing of $\triangle x$ can lead to instability of the numerical scheme in question. The notation of stability will be the topic of the next section.